Find the volume $ V $ of a pyramid if side $a = 23$ and slant height $s = 24$.

$V = \frac{ 2645 \sqrt{ 71}}{ 6 }$

STEP 1: find height $ h $

To find height $ h $ use Pythagorean Theorem:

$$ h^2 + \frac{ a^2 }{ 4 } = s^2 $$

After substituting $ a = 23 $ and $ s = 24 $ we have:

$$ h ^ {\,2} + \frac{ 23^2 }{ 4 } = 24^2 $$ $$ h ^ {\,2} + \frac{ 529 }{ 4 } = 24^2 $$ $$ h ^ {\,2} + \frac{ 529 }{ 4 } = 24^2 $$ $$ h ^ {\,2} = 576 - \frac{ 529 }{ 4 } $$ $$ h ^ {\,2} = \frac{ 1775 }{ 4 } $$ $$ h = \sqrt{ \frac{ 1775 }{ 4 } } $$$$ h = \frac{ 5 \sqrt{ 71}}{ 2 } $$

STEP 2: find volume $ V $

To find volume $ V $ use formula:

$$ V = \dfrac{ a ^{ 2 } \cdot h }{ 3 } $$

After substituting $ a = 23 $ and $ h = \frac{ 5 \sqrt{ 71}}{ 2 } $ we have:

$$ V = \dfrac{ 529 \cdot \frac{ 5 \sqrt{ 71}}{ 2 } }{ 3 }$$$$ V = \dfrac{ \frac{ 2645 \sqrt{ 71}}{ 2 } }{ 3 } $$$$ V = \frac{ 2645 \sqrt{ 71}}{ 6 } $$

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